For the eggplant:
Heat the oven to 350°F and arrange a rack in the middle.
Cut the eggplant(s) lengthwise into 1/8-inch-thick slices. Place in a single layer on a flat surface or 2 baking sheets, overlapping slightly as needed, and sprinkle evenly with 1 teaspoon of the salt. Flip the eggplant and sprinkle evenly with the remaining 1 teaspoon salt; let sit until water beads form on the surface, at least 30 minutes. Meanwhile, prepare the sauce.
For the sauce:
Using a food processor fitted with a blade attachment, pulse the tomatoes and their juices, in batches as needed, until coarsely chopped (about 10 pulses). Heat the oil in a large saucepan with a tightfitting lid over medium-high heat until shimmering. Add the onion and cook, stirring occasionally, until translucent, about 3 minutes. Add the garlic and cook until fragrant, about 30 seconds more.
Push the onions and garlic to one side of the pan and add the tomato paste to the empty side of the pan. Cook the paste slightly to remove the raw flavor, stirring occasionally, about 1 to 2 minutes. Stir the onions and garlic into the paste to incorporate. Add the chopped tomatoes, bay leaf, red pepper flakes, and a few pinches of salt. Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for 45 minutes to meld the flavors.
Add the capers, taste, and season with additional salt and red pepper flakes as needed; set aside.
To finish the eggplant:
Using paper towels, pat the eggplant slices dry on both sides. In a large nonstick frying pan, heat 1 1/2 teaspoons of the oil over medium-high heat. Add just enough eggplant to sit in a single layer in the pan and sear on both sides, about 4 minutes total. Taste and season with salt and pepper as needed. Transfer to a plate and repeat, in batches, with another 1 1/2 tablespoons of the oil and the remaining uncooked eggplant.
While the eggplant cooks, place the remaining 2 tablespoons olive oil, parsley, vinegar, red pepper flakes, and a pinch of salt in a large bowl and stir to combine. Transfer the seared eggplant to the oil-vinegar mixture and toss. Taste and season with additional salt as needed.
For the noodles:
Bring a large pot of heavily salted water to a boil over medium-high heat. Add the noodles and cook, stirring occasionally, until al dente, about 7 minutes. Drain. When cool enough to handle, lay the pieces flat on a lightly oiled baking sheet.
For the filling:
Place the tofu, parsley, nutritional yeast (if using), lemon zest, lemon juice, and measured salt and pepper in the bowl of a food processor fitted with a blade attachment and process until smooth, about 30 seconds. Taste and season with more lemon juice, salt, and pepper as needed; set aside.
To assemble the lasagna:
Spread a thin layer of tomato sauce in the bottom of a 13-by-9-inch baking dish. Place a single layer of noodles on top of the sauce, about 3 regular-sized noodles. Top the noodles with a quarter of the tofu filling (about 1 cup) and spread evenly. Lay a quarter of the eggplant slices over the filling. Spread about 1 cup of sauce over the eggplant and sprinkle with about 1/4 cup of the basil leaves. Make three more layers of noodles, filling, eggplant, sauce, and basil, omitting the basil from the top layer.
Cover with foil and bake for 50 minutes. Uncover and bake until bubbling, about 10 minutes more. Let cool at least 10 minutes before cutting. Sprinkle with the remaining 1/4 cup basil. Serve with any remaining tomato sauce.